7 ideas
13489 | Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD] |
Full Idea: Von Neumann's decision was to start with the ordinals and to treat cardinals as a special sort of ordinal. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by William D. Hart - The Evolution of Logic 3 | |
A reaction: [see Hart 73-74 for an explication of this] |
12336 | A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou] |
Full Idea: In Von Neumann's definition an ordinal is a transitive set in which all of the elements are transitive. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Alain Badiou - Briefings on Existence 11 |
18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy] |
Full Idea: For Von Neumann the successor of n is n U {n} (rather than Zermelo's successor, which is {n}). | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 |
18180 | Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann] |
Full Idea: Von Neumann's version of the natural numbers is in fact preferred because it carries over directly to the transfinite ordinals. | |
From: comment on John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n9 |
15925 | Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine] |
Full Idea: Each Von Neumann ordinal number is the set of its predecessors. ...He had shown how to introduce ordinal numbers as sets, making it possible to use them without leaving the domain of sets. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Shaughan Lavine - Understanding the Infinite V.3 |
9591 | The human intellect has not been, and cannot be, fully formalized [Nagel/Newman] |
Full Idea: The resources of the human intellect have not been, and cannot be, fully formalized. | |
From: E Nagel / JR Newman (Gödel's Proof [1958], VIII) | |
A reaction: This conclusion derives from Gödel's Theorem. Some people (e.g. Penrose) get over-excited by this discovery, and conclude that the human mind is supernatural. Imagination is the key - it is a feature of rationality that escapes mechanization. |
16764 | The soul conserves the body, as we see by its dissolution when the soul leaves [Toletus] |
Full Idea: Every accident of a living thing, as well as all its organs and temperaments and its dispositions are conserved by the soul. We see this from experience, since when that soul recedes, all these dissolve and become corrupted. | |
From: Franciscus Toletus (Commentary on 'De Anima' [1572], II.1.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.5 | |
A reaction: A nice example of observing a phenemonon, but not being able to observe the dependence relation the right way round. Compare Descartes in Idea 16763. |